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Shear capacity of 3D composite CFT joints subjected to symmetric loading condition
Journal of Constructional Steel Research 112 (2015) 242–251
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Journal of Constructional Steel Research
Shear capacity of 3D composite CFT joints subjected to symmetric loading condition Jiansheng Fan a,⁎, Cheng Liu a, Yue Yang a, Yu Bai b, Chao Wu b a b
Department of Civil Engineering, Tsinghua University, Beijing 100084, China Department of Civil Engineering, Monash University, Melbourne, Australia
a r t i c l e
i n f o
Article history: Received 30 December 2014 Accepted 8 May 2015 Available online 5 June 2015 Keywords: Composite joint 3D joint Panel zone Shear capacity CFT column
a b s t r a c t Shear capacity analysis of the panel zone in a composite joint of concrete-filled steel tubular (CFT) column and steel beam is important for avoidance of premature shear failure of the joint. This paper reviews 2D shear capacity models for joints between CFT column and steel beam. Then a 3D model is proposed for consideration of joints subjected to symmetric beam loadings in two planes and compared to the existing 2D shear capacity models. The effects of encased concrete on the shear capacity of the joint are taken into account for 3D composite joints through an additional compression strut model. A shear force and deformation relationship with four linear segments is thus achieved for a composite joint. To evaluate the corresponding ultimate shear capacity, the modeling results are compared with experiments where specimens fail through shear mode at the joints. It is found that the shear–deformation relationship and ultimate shear capacities predicted for 3D composite joints agree well with the experimental results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction A concrete-filled steel tubular (CFT) column is a structural element in which steel tube and encased concrete carry load by composite action between them. The strength and ductility of the encased concrete are improved by the confinement effect of the steel tube while the local buckling of steel tube is delayed by the encased concrete. Due to advantages such as high strength, excellent ductility and convenience for production and construction, CFT columns are widely used in civil infrastructure. In high-rise buildings, a typical structure system is formed by CFT columns connecting with horizontal structural members (such as beams and floor panels). To ensure structural ductility, premature shear failure of a composite joint should be avoided, especially in structural seismic design. Resistance to earthquake loading in such structural systems depends largely on the capacity of beam-to-column composite joints. Therefore, reliable estimation of the mechanical performance of composite joints between CFT column and beam is essential for structural design. Extensive experimental and theoretical studies have been performed to understand the performance of the panel zone in a composite joint between CFT column and steel beam and to develop corresponding mechanical models. Analytic formulations for joint shear capacity have also been developed [1–5], but only a 2D configuration (i.e. with CFT column and steel beam in the same plane) has
⁎ Corresponding author. E-mail address: [email protected] (J. Fan).
http://dx.doi.org/10.1016/j.jcsr.2015.05.005 0143-974X/© 2015 Elsevier Ltd. All rights reserved.
been considered. Research has shown that for a composite joint with internal diaphragms, the steel plates in the joint region and the encased concrete both contribute to the shear capacity of the joint. Using the superposition principle, equations have been formulated to estimate the shear capacity of 2D composite joints by the Architectural Institute of Japan (AIJ) [6]. In the latter research, however, the effects of axial compression on the shear capacity of the panel zone were not considered. Also, it was required that the nominal compression strength of the encased concrete should not be higher than 36 MPa, which largely limited its applicability for high-strength concrete. Koester [1] conducted a series of experiments on split-tee through-bolted moment joints between CFT columns and wide-flange steel beams. In this configuration, the shear capacity of the panel zone of the composite joints was estimated through regression analysis of experimental results without mechanism-based modeling. This limitation therefore restricts the applicability of the proposed shear capacity equation for other forms of composite joints with different connection configurations. Cheng et al. [2,3] proposed an innovative stress–strain model for the panel zone of a composite joint based on the compression strut mechanism, in which the shear force transfer was taken into account through a truss mechanism. The shear capacity formulation developed in this way considered the effects of axial compression on the steel wall and the encased concrete of the column; however, the model became complicated and was not convenient for design purposes. Nishiyama et al. [4] carried out a series of experiments on beam-to-column joints of CFT columns made of high-strength steel and concrete. Their experimental results showed that the design formula given in AIJ [6] was applicable for unconfined compression strength of concrete up to 110 MPa
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2. Analytic shear capacity model of a 3D composite joint
(a)
Composite joints in building structures are often under 3D loading conditions, such as those for interior columns, exterior columns and corner columns. In a CFT column system connected by orthogonal beams, the core zone of the 3D composite joint is actually subjected to shear forces from two orthogonal planes, the mechanism of which may be different from that of a planar composite joint. In this section, we establish analytic models of the shear–deformation relationship of steel tube and concrete core. Classical plastic theory is applied to derive the yield and ultimate shear strength of the steel tube. Two forms of strut mechanism are considered to model the ultimate shear capacity of the concrete core. The joint shear–deformation relationship is obtained by the superposition principle, considering the contributions from both steel webs and concrete core. It should be noted that the 3D loading conditions considered in this paper are limited to the scenario where the shear forces from two orthogonal directions are equal (resultant shear force in 45° direction) and the planar shape of the concrete panel is square, since this scenario is the most typical for 3D loading conditions.
(b)
Fig. 1. Distribution of shear stress (τ) along webs of a steel tube in (a) elastic and (b) plastic stage under bi-directional loading (Vx = Vy).
and tensile strength of steel up to 809 MPa. Similar experiments were conducted by Fukumoto and Morita [5] on the joints of high-strength CFT columns and steel beams to investigate structural elastoplastic behavior. A new compression strut mechanism was proposed, where a trilinear shear–deformation relationship was derived for description of the full shear deformation of the panel zone, and the effects of axial force on the behavior of the steel tube were also considered. In that study, the shear deformation at the ultimate strength of the encased concrete was defined as the ultimate shear deformation of the composite joint. Since the ultimate shear strain of concrete is much lower than that of a steel tube, the shear capacity at yielding of the webs of the steel tube was considered in the calculation of the ultimate shear capacity of the joint. In another words, the shear capacity of the joint was provided by the yielding strength of the steel tube plus the ultimate strength of the encased concrete. Existing formulations of joint shear capacity only consider the shear forces in one plane, corresponding to a 2D composite joint configuration. For realistic composite joints under seismic action, the joint panel zone is actually subjected to shear forces in two planes, corresponding to a 3D joint configuration. For such cases, no mechanical modeling has yet been done to describe joint stiffness and to predict joint shear capacity. In this paper, the 2D load transfer mechanism is extended to a 3D construct to analyze the shear capacity of composite joints in CFT column systems. A 3D load transfer mechanism is analytically modeled and the contributions from both steel column and encased concrete to the shear capacity of the joint panel zone are investigated. In this 3D model, a shear–deformation relationship with four linear segments is considered and its reliability and applicability are verified against previous experimental results including scenarios of composite joints with normal strength concrete [7] and with highstrength concrete [4].
(a)
2.1. Shear capacity and deformation of steel tube at joint region The shear capacity of a steel tube at the joint region has two components. One component derives from the shear strength of the webs of the steel tube and the other derives from the shear capacity of the steel tube–inner diaphragm system. In 3D composite joints, the latter component derives mainly from the deformation of the diaphragm system from cubic to parallelepiped shape (similar to a frame mechanism) and it has been reported that this contribution to the total shear capacity of the steel tube at the joint region is very limited [8]. Therefore, only the contribution of the webs to the overall shear capacity of the steel tube is taken into account in this study. In 3D joint panel zones subjected to shear forces from both x and y directions, the shear stresses in the webs of the steel tube can be determined by principles of material mechanics. Fig. 1 shows the distributions of shear stresses along the webs of steel tube in the elastic (Fig. 1a) and plastic stage (Fig. 1b) under bidirectional loading. If Vx = Vy, the resultant shear force V acts in a 45° direction to Vx or Vy. In the elastic stage, the maximum shear stress within the webs of the steel tube (see Fig. 1(a)) can be calculated as: τmax ¼
κ sV Aw
ð1Þ
where τmax is the maximum shear stress within the webs of the steel tube; Aw is the shear area of the webs of the steel tube; κs is the shear coefficient, which equals 1.2 for a square hollow section according to Fukumoto and Morita [5].
(c)
(c)
Fig. 2. Compression strut model for concrete at panel zone of a composite joint under loading in (a) XZ plane, (b) YZ plane and (c) 3D loading (bi-directional loading).
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251
Assuming a uniform shear stress distribution in the webs of the steel tube in the plastic stage (Fig. 1(b)), the shear stress can then be calculated as: V τ¼ Aw
ð2Þ
where τ is the shear stress of the webs of the steel tube. According to the von Mises criterion, the yield (τy) and maximum (τu) shear stress sustained by the webs of the steel tube can be calculated respectively as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f sy −σ 2s pffiffiffi τy ¼ 3
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 τy Aw Aw f sy −σ s pffiffiffi ¼ ¼ ks 3
V su ¼ τu Aw ¼
θ3D-Eq.(15)
25
θ3D-Eq.(16) θ2D
20 15 10 5 0.5
where fsy and fsu are the yielding and ultimate strength of the steel tube respectively and σs is the axial compression stress of the steel tube. Hence the yielding (Vsy) and ultimate (Vsu) shear capacities of the webs can be obtained as:
Aw
30
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f su −σ 2s pffiffiffi τu ¼ 3
V sy
35
Inclination Angle θ /°
244
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f su −σ 2s pffiffiffi : 3
ð5Þ
ð6Þ
Eqs. (5) and (6) suggest that the shear capacity of steel webs under bi-directional 3D loading is the same as their shear capacity under 2D loading. The yielding (γsy) and ultimate (γsu) shear deformations of the steel tube of a plain joint are given as [5]:
γ sy ¼ κ s
V sy Aw Gs
ð7Þ
γ su ¼ κ s
V su −V sy þ γsy Aw G0s
ð8Þ
1.0
1.5
2.0
Aspect Ratio α
2.5
3.0
Fig. 4. Relationship between inclination angle θ and aspect ratio α under 3D and planar loading states.
where the shear modulus of the steel tube Gs is normally 79 GPa. Gs′ is the slope of the second stage of the trilinear shear–deformation relationship of the steel tube and can be expressed as [5]: G0s ¼
1 1 9 þ Gs α s Es σ 2s =τ0 2 þ 3
where τ 0 ¼ p1ffiffi3
f sy þ f su ; 2
ð9Þ
αs is the ratio between the tangent modulus of
the second stage and that of the first stage in the trilinear shear– deformation relationship and can be taken as 0.1 according to material testing results [7]. 2.2. Shear capacity and deformation of concrete at panel zone As in a 2D joint, the shear transfer mechanism of the concrete at the panel zone of a 3D joint can be described using the strut model [9]. In this model, two mechanisms – the shear capacities of the main compression strut (Vcu1) and the additional confined compression strut (Vcu2) – contribute to the overall shear capacity of the concrete. For a joint under planar loading, the main compression strut and the additional confined compression strut are illustrated in Fig. 2(a) for loading in the XZ plane and in Fig. 2(b) for loading in the YZ plane. The corresponding shear–deformation relationship was detailed by Fukumoto and Morita [5]. 0.35
Vcu1-3D
0.30
Vcu1-2D
Vcu1/σcbdc2
0.25
Vcu1-x
0.20 0.15 0.10 0.05 0.5
Fig. 3. Stress state of main compression strut mechanism of concrete in the panel zone of a 3D composite joint.
1.0
1.5
2.0
Aspect Ratio α
2.5
3.0
Fig. 5. Relationship between shear capacity of main strut Vcu1/σcbd2c and aspect ratio α under 3D and planar loading states.
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Considering the force equilibrium in the horizontal direction of the isolated body (Fig. 3), the shear capacity of the main compression strut of concrete Vcu1-3D can be established as: 2
V cu1‐3D ¼ σ cb b sin θ3D cos θ3D
ð10Þ
where σcb is the compression strength of the concrete; θ3D is the inclination angle of the compression strut with respect to the square concrete core [5]; b is a dimension parameter illustrated in Fig. 3 and is determined by geometric relation as: pffiffiffi 2 h tan θ 2
b ¼ dc −
where dc and h are the width and the height respectively of the panel zone as illustrated in Fig. 3. Substituting Eq. (11) into Eq. (10) gives:
Fig. 6. Additional compression struts of concrete in the panel zone of a 3D joint.
2
Moreover, with the loadings from two planes in a 3D joint, the main and additional compression struts as shown in Fig. 2(c) are in 3D stress states. Accordingly, their orientation, shape and size may differ from those in a planar loading state. According to theory developed for reinforced concrete, the shear capacity of the main compression strut is formed through an arch mechanism. The additional compression struts in a 3D composite joint (Fig. 2(c)) are further confined by the steel tube flange. The shapes and orientations of the main and additional struts in the 3D composite joint shown in Fig. 2(c) were obtained as a result of the superposition effect of the corresponding main and additional struts from the two planar loading scenarios (i.e. Fig. 2(a) and (b)). Hence the main strut in a 3D state is determined as the overlap of the two main struts from two planar loading states in Fig. 2(a) and (b). To extend the modeling of concrete shear capacity of a 2D joint to a 3D configuration, the challenge is to take into account the 3D stress states of the main and additional compression struts. The subscripts “3D” and “2D” are used below to distinguish the variables (e.g. shear capacities) in 3D and planar loading states. The mechanism of the main compression strut in a 3D shear state is illustrated in Fig. 3. Vcu1-3D is the shear capacity of the main strut and σcbb2cosθ3D is the corresponding axial force. θ3D is the inclination angle of the main strut with respect to the concrete core (i.e. the angle between AC1 and axis Z). Plane C1C2C3C4 presents a cross section cut from the main compression strut to form an isolated body for establishing the relationship between the shear capacity Vcu1-3D and the axial force σcbb2cos θ3D.
(a)
ð11Þ
(b)
!2 pffiffiffi 2 α tan θ3D sin θ3D cos θ3D 2
V cu1‐3D ¼ σ cb dc 1−
ð12Þ
where α is the aspect ratio of the panel zone, defined as:
α¼
h : dc
ð13Þ
From Eq. (12) it can be seen that the inclination angle θ3D should be determined first, in order to calculate the shear capacity of main compression strut (Vcu1-3D). Based on the lower bound theorem of classical plasticity theory [10], the stress state illustrated in Fig. 3 is a statically admissible field and the corresponding shear capacity is the lower bound of the true shear capacity. The maximum value of Vcu1-3D with respect to θ3D in Eq. (12) represents the closest estimation of the true shear capacity. Therefore, the inclination angle θ3D can be determined when Vcu1-3D reaches its maximum value in Eq. (12). In the meantime, θ3D must satisfy the geometrical constraints of the main strut given in pffiffiffi Eq. (14), i.e. tanθ3D should be less than 2 dc/h otherwise point C1 will exceed the bounds of concrete core (see Fig. 3).
0b tan θ3D b
pffiffiffi 2 α
ð14Þ
(c)
Fig. 7. Derivation of the shear capacity of the additional compression strut for a 3D joint (a) horizontal force equilibrium, (b) ultimate limit state of steel flanges, and (c) axial force equilibrium of additional strut.
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To find the maximum point of Vcu1-3D, Eq. (12) is differentiated with respect to θ3D. Eq. (15) can be obtained therefore:
ð15Þ Eq. (15) is a nonlinear equation that determines the relationship between the aspect ratio α and inclination angle θ3D. It is difficult to find the analytical solution of inclination angle θ3D as a function of aspect ratio α. Matlab 2009b® was used to obtain a numerical solution for various values of aspect ratio α in the range of engineering practice (from 0.5 to 3) as shown in Fig. 4. As shown in Fig. 4, the inclination angle θ3D decreases with the increase of aspect ratio α in the range from 0.5 to 3. In this range, a simplified formulation for θ3D as a function of α is obtained by data regression analysis as: θ3D
dc 1 : ¼ 0:468 arctan ¼ 0:468 arctan α h
ð16Þ
The simplified formulation in Eq. (16) agrees well with the numerical results based on Eq. (15) as depicted in Fig. 4. Furthermore, the θ2D versus α curve was also derived for planar shear states by Fukumoto and Morita [5] and is illustrated in Fig. 4. It can be seen from Fig. 4 that, for a certain aspect ratio, the inclination angle in the planar shear state (θ2D) is always greater than that in the 3D shear state (θ3D). The shear capacity of the main compression strut can then be rewritten as: V cu1‐3D ¼
1 2 σ b sin2θ3D 2 cb
ð17Þ
where b and θ3D are determined by Eqs. (11) and (16) respectively. The x or y component (see Fig. 3) of shear capacity of the main compression strut in a 3D state can be calculated as: V cu1‐x ¼ V cu1‐y
pffiffiffi pffiffiffi 2 2 2 V cu1‐3D ¼ σ b sin2 θ3D : ¼ 2 4 cb
ð18Þ
The shear capacities under 3D and planar loading states (calculated according to Eqs. (17) to (18)) are illustrated in Fig. 5. It is indicated that the x component of shear capacity in a 3D shear state is always smaller than that in a planar shear state except for very large aspect ratios. This means that the spatial coupling effect weakens the shear
Vcu-x/Vcu-2D
pffiffiffi 2 " pffiffiffi ! # α tan θ3D −1 pffiffiffi 2 ∂V cu1‐3D α sin θ3D − cos θ3D cos2 θ3D þ 2α sin θ3D ¼ 0: ¼ 2 2 cos θ3D ∂θ3D
1.0
0.8
rw =20 rw =40
0.6
rw =60 rw =80
0.4
0
1
2
3
Aspect Ratio α =h/dc Fig. 9. Relation between rw and Vcu-x/Vcu-2D for rσ = 10.
capacity of the main compression strut in one plane, and this reduction becomes more notable for smaller aspect ratios. In a similar manner to the additional compression strut mechanism in a planar shear state [5], the additional compression struts in a 3D shear state are illustrated in Fig. 6. Four additional compression struts exist, as shown in Fig. 6 (with different colors), and they are confined by the flanges of the steel tube. Due to the interaction of the steel tube flange, two plastic hinges may be formed at the two ends of each additional compression strut. The formulation of the shear capacity Vcu2-3D of one additional strut is derived according to the mechanism illustrated in Fig. 7(a), (b), and (c). Because the x or y components of Vcu2-3D are provided by two additional compression struts, the contribution of each additional strut to pffiffiffi the overall shear capacity is obtained as 2Vcu2-3D/4. Considering the horizontal force equilibrium of each additional strut as shown in Fig. 7(a), Eq. (19) is established: pffiffiffi 2 V cu2‐3D ¼ pbla 4
ð19Þ
where p is the pressure from the flange of the steel tube (see Fig. 7(a)); b is a dimension parameter determined by geometric relation (Eq. (11)); la is the distance between two plastic hinges formed within the steel tube flanges and can be determined by following Eq. (20): 1 2 pbla dψ ¼ 2M f dψ 2
ð20Þ
0.60
1.0
Ratio of Vcu2-x andVcu2-2D Ratio
Vcu-x/Vcu-2D
0.58
0.56
rσ=5
0.8
rσ=10 rσ=15
0.6
rσ=20 0.4
0.54 0.5
1.0
1.5
2.0
2.5
3.0
Aspect Ratio α Fig. 8. Ratio of x (or y) component of 3D shear capacity to planar shear capacity of an additional strut as a function of aspect ratio α.
0
1
2
Aspect Ratio α =h/dc
Fig. 10. Relation between rσ and Vcu-x/Vcu-2D for rw = 40.
3
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247
V3D-x/V2D
1.0
rw=20 rw=40
0.8
rw=60 rw=80 0.6
0
1
2
3 Fig. 13. Shear–deformation relationship with four linear segments for the panel zone.
Aspect Ratio α =h/dc Fig. 11. Relationship between rw and V3D-x/V2D for rσ = 10.
where dΨ is an arbitrary small virtual rotational angle; Mf is the plastic moment capacity of the flange plate of the steel tube and can be expressed as: Mf ¼
1 2 bt f 4 s sy
ð21Þ
where b is a dimension parameter determined by geometric relation (Eq. (11)); ts is the thickness of the plate of the steel tube flange; and fsy is the material yield strength of the steel tube. Eq. (20) was derived with consideration of the ultimate limit state of the steel tube flange (see Fig. 7(b)). The virtual work of the pressure from the additional strut should equal to the virtual work of the plastic hinges of the steel flanges because of the principle of virtual work. In the ultimate limit state, the compression stress in the additional strut reaches the compression strength of concrete (see Fig. 7(c)), and Eq. (22) can therefore be obtained as: σ cb cos2 ϕ ¼ p
ð22Þ
where ϕ is the dihedral angle between the plane of the steel tube flange and the plane of the cross section of the additional strut (see Fig. 7(c)). ϕ is determined by the geometric relationship as: cos2 ϕ ¼
1 sin2 θ3D 2
ð23Þ
V3D-x/V2D
1.0
rσ=5 rσ=10
0.8
rσ=15
where the inclination angle θ3D of the main strut can be calculated by Eq. (16). Finally, the shear capacity of the additional strut in a 3D shear state is derived by combining Eqs. (19) to (23) as below: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V cu2‐3D ¼ 4 M f bσ cb sin θ3D :
ð24Þ
The x or y component (see Fig. 6) of shear capacity of the additional compression strut in a 3D state is actually: V cu2‐x ¼ V cu2‐y ¼
pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V cu2‐3D ¼ 2 2 M f bσ cb sin θ3D : 2
ð25Þ
The shear capacity of the additional strut in a planar shear state takes the form of Eq. (26) [5]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V cu2‐2D ¼ 4 M0f dc σ cb sin θ2D
ð26Þ
where dc is the width of the panel zone; σcb is the compression strength of the concrete; θ2D is the inclination angle of the compression strut with respect to the square concrete core in a planar shear state; Mf′ is the plastic moment capacity of the flange plate in a planar shear state and has a similar form to Eq. (21) except that b is replaced by dc: M0f ¼
1 2 dc t f : 4 s sy
ð27Þ
Since the inclination angle (θ2D) in a planar shear state is always larger than that in a 3D shear state (θ3D), and dc is also always larger than b (see Eq. (11)), by comparing Eqs. (24) and (26) it can be concluded that the 3D shear capacity of the additional strut is always lower than the planar shear capacity, and the x or y component of its 3D shear capacity is even smaller than the shear capacity of the additional strut in a planar shear state. This indicates that the spatial coupling effect also weakens the shear capacity of the additional compression struts. The ratio of the x (or y) component of 3D shear capacity Vcu2-x to the planar shear capacity Vcu2-3D is shown in Fig. 8 as a function of the aspect ratio
rσ=20 0.6
Table 1 Coefficients C1 and C2 for calculation of yielding shear capacity Vpy for a 3D composite joint used in Eq. (33).
0
1
2
Aspect Ratio α =h/dc
Fig. 12. Relationship between rσ and V3D-x/V2D for rw = 40.
3
Relationship of shear strains
γpy
γpu
C1
C2
γsy b γcu ‐ x b γsu γcu ‐ x b γsy b γsu γsy b γsu b γcu ‐ x
γsy γcu ‐ x γsy
γsu γsu γcu ‐ x
1 γcu ‐ x/γsy 1
γsy/γcu ‐ x 1 γsy/γcu ‐ x
248
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251
rewritten as Eq. (29) using the steel to concrete strength ratio rσ and width to thickness ratio rw of the steel tube:
V cu‐x ¼
!2 !3 pffiffiffi2 pffiffiffi pffiffiffi pffiffiffiffiffi 2 rσ 24 2 2 2 1− α tan θ3D 5 sin θ3D σ cb dc 1− α tan θ3D cos θ3D þ 2 2 2 rw
ð29Þ
Fig. 14. Dimensions of the joint specimens (J202).
α. It can be seen from Fig. 8 that a larger respect ratio tends to induce a more significant spatial coupling effect and therefore causes more reduction in the shear capacity of the additional compression strut in a 3D shear state. Considering that shear capacity is often designed in the directions of the principal axes (x and y) of the two orthogonal beams, the shear capacity (Vcu-x or Vcu-y) of the concrete in the panel zone of a 3D composite joint can be calculated in these two directions according to Eq. (28), which was derived based on Eqs. (17) and (24):
V cu‐x ¼
sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffi pffiffiffi 2 2 Mf b cos θ3D þ 4 ðV cu1‐3D þ V cu2‐3D Þ ¼ bσ cb sin θ3D : 2 2 bσ cb ð28Þ
To investigate the effects of major design parameters on the reduction of shear capacity due to the spatial coupling effect, Eq. (28) is
where the steel to concrete strength ratio rσ equals fsy/σcb and the width to thickness ratio of the steel tube rw equals dc/ts. It can be seen that for specified σcbd2c , the aspect ratio α, the strength ratio rσ and the width to thickness ratio of the steel tube rw are the factors that influence the shear capacity of the concrete panel. Figs. 9 and 10 show the influence of the aspect ratio (α) on the ratio of the shear capacity of the concrete core in a 3D loading state and in a 2D loading state (Vcu-x/Vcu-2D) with different width to thickness ratios (rw) and strength ratios (rσ). Both figures show that the aspect ratio α has remarkable influence on Vcu-x/Vcu-2D and a smaller aspect ratio α corresponds to a more significant reduction in the value of Vcu-x/Vcu-2D due to the spatial coupling effect. The shear capacity of the concrete core in a 3D loading state is only 50% of that in a 2D loading state with a small aspect ratio (0.5 b α b 1). This means that the spatial coupling effect cannot be ignored for joints with small aspect ratios. Figs. 9 and 10 also show that the 3D shear capacity of the concrete core (Vcu-x ) decreases with a decrease in the width to thickness ratio or an increase in the strength ratio, although the influences of these two factors are not as significant as that of the aspect ratio. The deformation at the ultimate shear strength of a 3D concrete core is given by Fukumoto and Morita [5]:
γcu‐x ¼ κ c
V cu‐x α cu Ac Gc
ð30Þ
where κc is the shear coefficient of the concrete core (1.2 for square concrete); Gc is the shear modulus of the concrete core; Ac is the
Fig. 15. Schematic diagram of experimental setup, joint details, and loading configuration for specimen J202.
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251 Table 2 Summary of four joint specimens. Specimen no.
Loading
Concrete slab
Concrete encased in steel tube
Steel beams
J101 J202 J203 J301
Bi-directional Bi-directional Uni-directional Bi-directional
No Yes Yes Yes
Yes Yes Yes No
In two planes In two planes In one plane In two planes
cross-sectional area of the concrete core. αcu is the stiffness reduction ratio as given by Fukumoto and Morita [5]: α cu ¼ 0:00158σ cb þ 0:0411
h þ 0:086 dc
ð31Þ
where the compression strength of the concrete σcb is in units of N/mm. 2.3. Shear capacity and deformation of a 3D composite joint The total shear capacity V3D-x of a joint in a 3D loading state is the superposition of the shear capacity of the concrete core (Eq. (6)) and that of the steel webs (Eq. (28)):
V 3D‐x ¼ V su þ V cu‐x
sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffi 2 Mf b cos θ3D þ 4 ¼ bσ cb sin θ3D 2 bσ cb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Aw f su −σ 2s pffiffiffi þ : 3
ð32Þ
It is also important to evaluate the ratio of shear capacities in 3D and planar loading states (V3D-x/V2D). For such evaluation, a yield to tensile ratio (fsu/fsy) of 1.3 and an axial compression ratio (N0/(Acσcb + Asfsy), where N0 is column axial force, Ac and As are section areas of concrete and steel tube respectively) of 0.2 are adopted to calculate the joint shear capacity V3D-x (using Eq. (32)). Figs. 11 and 12 clearly illustrate the effects of the aspect ratio α, width to thickness ratio rw and strength ratio rσ on the reduction of joint shear capacity subjected to loading from 2D to 3D. Again, the aspect ratio shows significant reduction in capacity. Also, higher width to thickness ratios and lower strength ratios lead to more remarkable spatial coupling effects. The effects of the width to thickness ratio rw and the strength ratio rσ are more considerable on joint shear capacity with smaller aspect ratios. In the worst case (i.e., small aspect and strength ratios but a large width to thickness
249
ratio), joint shear capacity in 3D loading may correspond to only 60% of that in 2D loading. The shear–deformation relationship of a 3D composite joint may include four linear segments, as a result of the superposition of the trilinear model of the steel tube and the ideal elastoplastic model of encased concrete. This is shown in Fig. 13 where point A (with a shear deformation value of γsy, see Eq. (7)) corresponds to the yielding point of the panel zone; point B (with a shear deformation value of γcu-x, see Eq. (30)) is determined by the shear deformation at which the concrete core just reaches the ultimate shear strength; and point C (with a shear deformation value of γsu, see Eq. (8)) represents the ultimate shear strength of the panel zone corresponding to the ultimate state of the steel tube. Therefore, the total yielding (Vpy) and the ultimate (Vpu) shear capacity of a composite joint can be defined as: V py ¼ C 1 V sy þ C 2 V cu
ð33Þ
V pu ¼ V su þ V cu
ð34Þ
where Vsy and Vsu are the yielding and ultimate shear capacities of the steel tube in the joint region; Vcu-x is the ultimate shear capacity of the concrete in the panel zone of the joint; C1 and C2 are the coefficients that can be determined according to Table 1. In this table, γsy and γsu are the yielding and ultimate shear strains of the steel tube of a 3D joint (see Eqs. (7) and (8)); γcu-x is the ultimate shear strain of the concrete in the panel zone of a 3D joint (see Eq. (30)); γpy and γpu are the calculated yielding and ultimate shear strains of the joint region. For composite joints with fully developed shear strength and shear strain in the joint region, the webs of the steel tube yield first and then the concrete in the panel zone reaches its ultimate shear strain. Finally, the web of the steel tube reaches its ultimate state. For such joints, the relationship γsy b γcu-x b γsu is valid, i.e. C1 and C2 in Eq. (33) are determined according to the second row in Table 1. 3. Experimental validation To validate the analytical modeling developed above, four specimens (J101, J202, J203 and J301) were designed and tested under reversedcyclic loading [7]. The specimens were CFT columns with steel beams through interior diaphragm connections. Specimen J202 was the reference specimen, the dimensions of which are shown in Fig. 14 in detail. The geometries of specimens J101 and J301 were the same as that of J202, but no concrete slab was placed on top of the steel beam for J101, and no encased concrete was placed in the steel tubular column (but with a different tube thickness of 12 mm) for J301. Specimen
Fig. 16. Typical failure modes of the four specimens (a) fracture of steel tube (J101); (b) crushing of encased concrete (J202) and (c) overall deformation (J101).
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Table 3 Comparison of predicted shear capacities with experimental results. No.
Vsu/kN
Vcu/kN
Vpu/kN
Veu/kN
J101 J202 J203 R1 R6
1336.0 1336.0 1336.0 806.4 808.7
284.0 275.4 457.2 1579.6 914.8
1620.0 1611.4 1793.2 2386.0 1723.5
1743.7 1594.8 1944.0 2384.6 1787.6
J203 had beams in only one plane and was tested under a uni-directional reversal loading whereas the other specimens had beams in two planes and were tested under bi-directional reversal loading. The experimental setup and loading configuration are shown in Fig. 15 and the key experimental parameters of the different joint specimens are listed in Table 2. Steel plates of 8 mm, 10 mm and 12 mm thickness were used to make the steel species; their measured yielding strengths fsy were 381.4 MPa, 345.9 MPa and 388.4 MPa, respectively. The measured compressive strengths (σcb) of the encased concrete in the columns and the concrete slabs were 51.1 MPa and 36.4 MPa respectively. During the tests, a constant axial force (2458 kN for J101, J202 and J301 and 2116 kN for J301) was first applied on the top of the column as shown in Figs. 14 and 15. Vertical reversed-cyclic loads were applied at the ends of four beams (two beams in each plane), in a force control mode until the measured strains in the steel beam or steel panel exceeded the material yield strain, and subsequently in a displacement control mode. The final failure mode of all four specimens was a typical shear failure in the panel zone of the joints, as shown in Fig. 16. The shear failures were characterized by fracture of the steel wall in the panel zone and crushing of the encased concrete within. These specimens therefore provide valuable information for examining the proposed analytical model developed for the joint shear capacity. According to the loading configuration of the specimens, the shear force Vj in the joint region could be calculated by Eq. (35) [11]: 2Q b Lb 2Q b Lb Q L −Q c ¼ − b H H b −t bf Hb −t bf
ð35Þ
Shear Force V/kN
where Qc is the vertical loading at the end of the beams and Qc is the horizontal reaction force at the top of the column; Hb is the height of the beam; tbf is the flange thickness of the beam; Lb is the distance from the loading point of the beam to the flange of the column; H is the joint height; L is the distance between the two loading points of the beams in the same plane (see Fig. 14). The shear capacities of the specimens were calculated using the proposed formulation developed for 3D composite joints (see Eq. (32)). The 2500
2500
2000
2000
1500
1500
1000 500
-3
γ /10
0 -40
-20
-500
0
20
40
-1000
experimental predicted
-1500 -2000 -2500
Shear Force V/kN
Vj ¼
calculated values are compared with the experimental results in Table 3, where Vsu and Vcu are the calculated contributions from the steel tube and the encased concrete according to Eqs. (6) and (28) respectively; Vpu is the total shear capacity of the 3D joint calculated by Eq. (32) and Veu is the measured shear capacity of the joint from experiments. Overall, good agreement between the calculated (Vpu) and measured (Veu) total shear capacities was found, with a maximum discrepancy less than 8%. Specimen J202 associated with beams in two planes was also compared to J203 with beams only in one plane. It was found that that the experimental result for the joint shear capacity of J202 was about 18% lower than that of J203, and 11% lower than the predicted shear capacity of J203. This comparison indicates that using the planar shear formulation may overestimate the joint shear capacity in the case of a 3D loading configuration and this may lead to unsafety. Further, joint shear–deformation curves were formed by the calculated shear capacity values (Vsy, Vsu and Vcu-x, according to Eqs. (5), (6) and (28)) and the corresponding shear deformation (γsy, γsu and γcu-x) according to Eqs. (7), (8) and (30) for specimen J202, as shown in Fig. 17. The highly nonlinear experimental shear–deformation skeleton curve can be relatively well described by the proposed modeling curve with four linear segments. Discrepancies between the experimental and predicted shear–deformation skeleton curves were mainly found at the second and third segments between the yield and ultimate states (see Fig. 13). These discrepancies may resulted from the assumption of an ideal elastoplastic shear–deformation relationship of the concrete core in Fig. 13. The 3D shear capacity model was further validated by the experimental results reported by Nishiyama et al. [4]. The specimens R1 and R6 [4] were tested under uni-directional and bi-directional reversal loading respectively, and were similar to specimens J202 and J101 investigated by [7]. The shear capacities of R1 and R6 were calculated using the shear capacity formulations developed in this paper and considering the material and dimensional parameters provided by Nishiyama et al. [4]. The experimental shear capacity was derived by Eq. (35) based on the force Qc (i.e. column shear force) applied at the end of the column [4]. The predicted and experimental results are compared in the last two rows in Table 3 and good agreement is evident, with discrepancies less than 5%. The planar shear capacity of specimen R6 was calculated as 2222.6 kN by the equations introduced by Nishiyama et al. [4] and this value was 24% higher than the experimental result (1787.6 kN, see Table 3). It is evident for specimen R6 that the spatial coupling effect significantly weakened the shear capacity of the joint subjected to 3D loading. This is because in this case, highstrength concrete (σcb = 97.7 MPa) was used for specimen R6, leading to a rather small steel to concrete strength ratio (rσ = fsy/σcb = 5.0). Together with the other factors (α = h/dc = 1.0, and rw = dc/ts = 55.6 according to Eq. (29)), this likely caused the more remarkable reduction to the shear capacity of the 3D joint.
1000 500
-3
γ /10
0 -40
-20
-500
0
-1000
20
40
experimental predicted
-1500 -2000 -2500
(a)
(b)
Fig. 17. Comparison of experimental and predicted skeleton curves of specimen (a) J202 (bi-directional loading) and (b) J203 (uni-directional loading).
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251
4. Conclusions An analytical model was developed in this paper to estimate the shear capacity of 3D composite joints of CFT columns and steel beams subjected to shear forces in two planes. In this model, the main compression strut and additional compression strut mechanisms were formulated for the concrete core of the joint panel subjected to 3D loading scenario. The joint shear capacity was calculated and compared with experimental results where the specimens were subjected to reversedcyclic loading in two planes and failed through shear mode at the joints. The joint shear–deformation relationship was characterized by a piecewise linear model with four segments, taking into consideration the effects of shear forces in two planes. The modeling results of joint shear capacities and shear–deformation relationships were validated by the experimental results from references. Based on this work, the following conclusions can be drawn: 1) The analytical model developed in this paper suggests that 3D (bidirectional) loading weakens the shear capacity of the concrete panel zone, although it does not affect the shear capacity of the webs of the steel tube. Because of the loadings from two planes and the associated space coupling effects on a 3D joint, the main and additional compression struts, obtained as a result of superposition of two 2D joints in the corresponding plane loading state, may present different orientations, shapes and sizes from those in a planar loading state. 2) A new piecewise linear constitutive model based on the superposition principle was implemented into the shear–deformation relationship of 3D composite joints. This constitutive model was characterized by the analytic ultimate shear strength formula in 3D shearing states, and therefore the spatial coupling effect caused by bi-directional shear to the concrete zone could be taken into consideration. The predicted and experimental shear–deformation curves showed satisfactory agreement. The major discrepancy was found between yield and ultimate states because of the assumption of an ideal elastoplastic shear–deformation relationship of the concrete core. 3) The proposed model for the shear capacity of composite joints of CFT columns was validated by the results from 2D and 3D joint experiments. Further calculations also indicated that using the planar shear capacity formulation for a joint in a 3D shear state may lead to an overestimation of 10%–20% for the joint shear capacity. The reduction in joint shear capacity due to a 3D loading configuration depends mainly on the aspect ratio of the joint panel, the steel to
251
concrete strength ratio and the width to thickness ratio of the steel tube. A decrease in the aspect ratio results in the most notable reduction in joint shear capacity and such a reduction also becomes more considerable with an increase in the width to thickness ratio or a decrease in the strength ratio. For a worst-case scenario where the joint has a relatively small aspect ratio (0.5 for example), a large width to thickness ratio (80.0 for example) and a small strength ratio (5.0 for example), joint shear capacity in the 3D shear state can reduce to only 60% of that in the 2D loading state.
Acknowledgment The authors appreciate the financial support of the National Natural Science Foundation of China (No. 51222810) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120002110003). References [1] B.D. Koester, Panel Zone Behavior of Moment Connections Between Rectangular Concrete-filled Steel Tubes and Wide Flange Beams, University of Texas at Austin, 2000. [2] C.-T. Cheng, L.-L. Chung, Seismic performance of steel beams to concrete-filled steel tubular column connections, J. Constr. Steel Res. 59 (2003) 405–426. [3] C.-T. Cheng, C.-F. Chan, L.-L. Chung, Seismic behavior of steel beams and CFT column moment-resisting connections with floor slabs, J. Constr. Steel Res. 63 (2007) 1479–1493. [4] I. Nishiyama, T. Fujimoto, T. Fukumoto, K. Yoshioka, Inelastic force–deformation response of joint shear panels in beam-column moment connections to concrete-filled tubes, J. Struct. Eng. 130 (2004) 244–252. [5] T. Fukumoto, K. Morita, Elastoplastic behavior of panel zone in steel beam-toconcrete filled steel tube column moment connections, J. Struct. Eng. 131 (2005) 1841–1853. [6] Architectural Institute of Japan (AIJ), AIJ Standard for Structural Calculation of Steel Reinforced Concrete Structures, 1987. (Tokyo, Japan). [7] J.S. Fan, Q.W. Li, J.G. Nie, H. Zhou, Experimental Study on the seismic performance of 3D joint between concrete-filled square steel tubular column and composite beams, J. Struct. Eng. (2013)http://dx.doi.org/10.1061/(ASCE)ST.1943-541X.0001013 (accepted for publication). [8] J.G. Nie, K. Qin, C.S. Cai, Seismic behavior of connections composed of CFSSTCs and steel–concrete composite beams—finite element analysis, J. Constr. Steel Res. 64 (2008) 680–688. [9] T. Paulay, R. Park, M.J.N. Preistley, Reinforced concrete beam-column joints under seismic actions, ACI Journal Proceedings, ACI, 1978. [10] W.-F. Chen, Limit Analysis and Soil Plasticity, Elsevier, 2013. [11] S. Sasaki, M. Teraoka, K. Morita, T. Fujiwara, Structural behavior of concrete-filled square tubular column with partial-penetration weld corner seam to steel Hbeam connections, Proceedings of the 4th Pacific Structural Steel Conference, 2 1995, pp. 33–40.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Shear capacity of 3D composite CFT joints subjected to symmetric loading condition Jiansheng Fan a,⁎, Cheng Liu a, Yue Yang a, Yu Bai b, Chao Wu b a b
Department of Civil Engineering, Tsinghua University, Beijing 100084, China Department of Civil Engineering, Monash University, Melbourne, Australia
a r t i c l e
i n f o
Article history: Received 30 December 2014 Accepted 8 May 2015 Available online 5 June 2015 Keywords: Composite joint 3D joint Panel zone Shear capacity CFT column
a b s t r a c t Shear capacity analysis of the panel zone in a composite joint of concrete-filled steel tubular (CFT) column and steel beam is important for avoidance of premature shear failure of the joint. This paper reviews 2D shear capacity models for joints between CFT column and steel beam. Then a 3D model is proposed for consideration of joints subjected to symmetric beam loadings in two planes and compared to the existing 2D shear capacity models. The effects of encased concrete on the shear capacity of the joint are taken into account for 3D composite joints through an additional compression strut model. A shear force and deformation relationship with four linear segments is thus achieved for a composite joint. To evaluate the corresponding ultimate shear capacity, the modeling results are compared with experiments where specimens fail through shear mode at the joints. It is found that the shear–deformation relationship and ultimate shear capacities predicted for 3D composite joints agree well with the experimental results. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction A concrete-filled steel tubular (CFT) column is a structural element in which steel tube and encased concrete carry load by composite action between them. The strength and ductility of the encased concrete are improved by the confinement effect of the steel tube while the local buckling of steel tube is delayed by the encased concrete. Due to advantages such as high strength, excellent ductility and convenience for production and construction, CFT columns are widely used in civil infrastructure. In high-rise buildings, a typical structure system is formed by CFT columns connecting with horizontal structural members (such as beams and floor panels). To ensure structural ductility, premature shear failure of a composite joint should be avoided, especially in structural seismic design. Resistance to earthquake loading in such structural systems depends largely on the capacity of beam-to-column composite joints. Therefore, reliable estimation of the mechanical performance of composite joints between CFT column and beam is essential for structural design. Extensive experimental and theoretical studies have been performed to understand the performance of the panel zone in a composite joint between CFT column and steel beam and to develop corresponding mechanical models. Analytic formulations for joint shear capacity have also been developed [1–5], but only a 2D configuration (i.e. with CFT column and steel beam in the same plane) has
⁎ Corresponding author. E-mail address: [email protected] (J. Fan).
http://dx.doi.org/10.1016/j.jcsr.2015.05.005 0143-974X/© 2015 Elsevier Ltd. All rights reserved.
been considered. Research has shown that for a composite joint with internal diaphragms, the steel plates in the joint region and the encased concrete both contribute to the shear capacity of the joint. Using the superposition principle, equations have been formulated to estimate the shear capacity of 2D composite joints by the Architectural Institute of Japan (AIJ) [6]. In the latter research, however, the effects of axial compression on the shear capacity of the panel zone were not considered. Also, it was required that the nominal compression strength of the encased concrete should not be higher than 36 MPa, which largely limited its applicability for high-strength concrete. Koester [1] conducted a series of experiments on split-tee through-bolted moment joints between CFT columns and wide-flange steel beams. In this configuration, the shear capacity of the panel zone of the composite joints was estimated through regression analysis of experimental results without mechanism-based modeling. This limitation therefore restricts the applicability of the proposed shear capacity equation for other forms of composite joints with different connection configurations. Cheng et al. [2,3] proposed an innovative stress–strain model for the panel zone of a composite joint based on the compression strut mechanism, in which the shear force transfer was taken into account through a truss mechanism. The shear capacity formulation developed in this way considered the effects of axial compression on the steel wall and the encased concrete of the column; however, the model became complicated and was not convenient for design purposes. Nishiyama et al. [4] carried out a series of experiments on beam-to-column joints of CFT columns made of high-strength steel and concrete. Their experimental results showed that the design formula given in AIJ [6] was applicable for unconfined compression strength of concrete up to 110 MPa
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251
243
2. Analytic shear capacity model of a 3D composite joint
(a)
Composite joints in building structures are often under 3D loading conditions, such as those for interior columns, exterior columns and corner columns. In a CFT column system connected by orthogonal beams, the core zone of the 3D composite joint is actually subjected to shear forces from two orthogonal planes, the mechanism of which may be different from that of a planar composite joint. In this section, we establish analytic models of the shear–deformation relationship of steel tube and concrete core. Classical plastic theory is applied to derive the yield and ultimate shear strength of the steel tube. Two forms of strut mechanism are considered to model the ultimate shear capacity of the concrete core. The joint shear–deformation relationship is obtained by the superposition principle, considering the contributions from both steel webs and concrete core. It should be noted that the 3D loading conditions considered in this paper are limited to the scenario where the shear forces from two orthogonal directions are equal (resultant shear force in 45° direction) and the planar shape of the concrete panel is square, since this scenario is the most typical for 3D loading conditions.
(b)
Fig. 1. Distribution of shear stress (τ) along webs of a steel tube in (a) elastic and (b) plastic stage under bi-directional loading (Vx = Vy).
and tensile strength of steel up to 809 MPa. Similar experiments were conducted by Fukumoto and Morita [5] on the joints of high-strength CFT columns and steel beams to investigate structural elastoplastic behavior. A new compression strut mechanism was proposed, where a trilinear shear–deformation relationship was derived for description of the full shear deformation of the panel zone, and the effects of axial force on the behavior of the steel tube were also considered. In that study, the shear deformation at the ultimate strength of the encased concrete was defined as the ultimate shear deformation of the composite joint. Since the ultimate shear strain of concrete is much lower than that of a steel tube, the shear capacity at yielding of the webs of the steel tube was considered in the calculation of the ultimate shear capacity of the joint. In another words, the shear capacity of the joint was provided by the yielding strength of the steel tube plus the ultimate strength of the encased concrete. Existing formulations of joint shear capacity only consider the shear forces in one plane, corresponding to a 2D composite joint configuration. For realistic composite joints under seismic action, the joint panel zone is actually subjected to shear forces in two planes, corresponding to a 3D joint configuration. For such cases, no mechanical modeling has yet been done to describe joint stiffness and to predict joint shear capacity. In this paper, the 2D load transfer mechanism is extended to a 3D construct to analyze the shear capacity of composite joints in CFT column systems. A 3D load transfer mechanism is analytically modeled and the contributions from both steel column and encased concrete to the shear capacity of the joint panel zone are investigated. In this 3D model, a shear–deformation relationship with four linear segments is considered and its reliability and applicability are verified against previous experimental results including scenarios of composite joints with normal strength concrete [7] and with highstrength concrete [4].
(a)
2.1. Shear capacity and deformation of steel tube at joint region The shear capacity of a steel tube at the joint region has two components. One component derives from the shear strength of the webs of the steel tube and the other derives from the shear capacity of the steel tube–inner diaphragm system. In 3D composite joints, the latter component derives mainly from the deformation of the diaphragm system from cubic to parallelepiped shape (similar to a frame mechanism) and it has been reported that this contribution to the total shear capacity of the steel tube at the joint region is very limited [8]. Therefore, only the contribution of the webs to the overall shear capacity of the steel tube is taken into account in this study. In 3D joint panel zones subjected to shear forces from both x and y directions, the shear stresses in the webs of the steel tube can be determined by principles of material mechanics. Fig. 1 shows the distributions of shear stresses along the webs of steel tube in the elastic (Fig. 1a) and plastic stage (Fig. 1b) under bidirectional loading. If Vx = Vy, the resultant shear force V acts in a 45° direction to Vx or Vy. In the elastic stage, the maximum shear stress within the webs of the steel tube (see Fig. 1(a)) can be calculated as: τmax ¼
κ sV Aw
ð1Þ
where τmax is the maximum shear stress within the webs of the steel tube; Aw is the shear area of the webs of the steel tube; κs is the shear coefficient, which equals 1.2 for a square hollow section according to Fukumoto and Morita [5].
(c)
(c)
Fig. 2. Compression strut model for concrete at panel zone of a composite joint under loading in (a) XZ plane, (b) YZ plane and (c) 3D loading (bi-directional loading).
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251
Assuming a uniform shear stress distribution in the webs of the steel tube in the plastic stage (Fig. 1(b)), the shear stress can then be calculated as: V τ¼ Aw
ð2Þ
where τ is the shear stress of the webs of the steel tube. According to the von Mises criterion, the yield (τy) and maximum (τu) shear stress sustained by the webs of the steel tube can be calculated respectively as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f sy −σ 2s pffiffiffi τy ¼ 3
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 τy Aw Aw f sy −σ s pffiffiffi ¼ ¼ ks 3
V su ¼ τu Aw ¼
θ3D-Eq.(15)
25
θ3D-Eq.(16) θ2D
20 15 10 5 0.5
where fsy and fsu are the yielding and ultimate strength of the steel tube respectively and σs is the axial compression stress of the steel tube. Hence the yielding (Vsy) and ultimate (Vsu) shear capacities of the webs can be obtained as:
Aw
30
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f su −σ 2s pffiffiffi τu ¼ 3
V sy
35
Inclination Angle θ /°
244
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f su −σ 2s pffiffiffi : 3
ð5Þ
ð6Þ
Eqs. (5) and (6) suggest that the shear capacity of steel webs under bi-directional 3D loading is the same as their shear capacity under 2D loading. The yielding (γsy) and ultimate (γsu) shear deformations of the steel tube of a plain joint are given as [5]:
γ sy ¼ κ s
V sy Aw Gs
ð7Þ
γ su ¼ κ s
V su −V sy þ γsy Aw G0s
ð8Þ
1.0
1.5
2.0
Aspect Ratio α
2.5
3.0
Fig. 4. Relationship between inclination angle θ and aspect ratio α under 3D and planar loading states.
where the shear modulus of the steel tube Gs is normally 79 GPa. Gs′ is the slope of the second stage of the trilinear shear–deformation relationship of the steel tube and can be expressed as [5]: G0s ¼
1 1 9 þ Gs α s Es σ 2s =τ0 2 þ 3
where τ 0 ¼ p1ffiffi3
f sy þ f su ; 2
ð9Þ
αs is the ratio between the tangent modulus of
the second stage and that of the first stage in the trilinear shear– deformation relationship and can be taken as 0.1 according to material testing results [7]. 2.2. Shear capacity and deformation of concrete at panel zone As in a 2D joint, the shear transfer mechanism of the concrete at the panel zone of a 3D joint can be described using the strut model [9]. In this model, two mechanisms – the shear capacities of the main compression strut (Vcu1) and the additional confined compression strut (Vcu2) – contribute to the overall shear capacity of the concrete. For a joint under planar loading, the main compression strut and the additional confined compression strut are illustrated in Fig. 2(a) for loading in the XZ plane and in Fig. 2(b) for loading in the YZ plane. The corresponding shear–deformation relationship was detailed by Fukumoto and Morita [5]. 0.35
Vcu1-3D
0.30
Vcu1-2D
Vcu1/σcbdc2
0.25
Vcu1-x
0.20 0.15 0.10 0.05 0.5
Fig. 3. Stress state of main compression strut mechanism of concrete in the panel zone of a 3D composite joint.
1.0
1.5
2.0
Aspect Ratio α
2.5
3.0
Fig. 5. Relationship between shear capacity of main strut Vcu1/σcbd2c and aspect ratio α under 3D and planar loading states.
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245
Considering the force equilibrium in the horizontal direction of the isolated body (Fig. 3), the shear capacity of the main compression strut of concrete Vcu1-3D can be established as: 2
V cu1‐3D ¼ σ cb b sin θ3D cos θ3D
ð10Þ
where σcb is the compression strength of the concrete; θ3D is the inclination angle of the compression strut with respect to the square concrete core [5]; b is a dimension parameter illustrated in Fig. 3 and is determined by geometric relation as: pffiffiffi 2 h tan θ 2
b ¼ dc −
where dc and h are the width and the height respectively of the panel zone as illustrated in Fig. 3. Substituting Eq. (11) into Eq. (10) gives:
Fig. 6. Additional compression struts of concrete in the panel zone of a 3D joint.
2
Moreover, with the loadings from two planes in a 3D joint, the main and additional compression struts as shown in Fig. 2(c) are in 3D stress states. Accordingly, their orientation, shape and size may differ from those in a planar loading state. According to theory developed for reinforced concrete, the shear capacity of the main compression strut is formed through an arch mechanism. The additional compression struts in a 3D composite joint (Fig. 2(c)) are further confined by the steel tube flange. The shapes and orientations of the main and additional struts in the 3D composite joint shown in Fig. 2(c) were obtained as a result of the superposition effect of the corresponding main and additional struts from the two planar loading scenarios (i.e. Fig. 2(a) and (b)). Hence the main strut in a 3D state is determined as the overlap of the two main struts from two planar loading states in Fig. 2(a) and (b). To extend the modeling of concrete shear capacity of a 2D joint to a 3D configuration, the challenge is to take into account the 3D stress states of the main and additional compression struts. The subscripts “3D” and “2D” are used below to distinguish the variables (e.g. shear capacities) in 3D and planar loading states. The mechanism of the main compression strut in a 3D shear state is illustrated in Fig. 3. Vcu1-3D is the shear capacity of the main strut and σcbb2cosθ3D is the corresponding axial force. θ3D is the inclination angle of the main strut with respect to the concrete core (i.e. the angle between AC1 and axis Z). Plane C1C2C3C4 presents a cross section cut from the main compression strut to form an isolated body for establishing the relationship between the shear capacity Vcu1-3D and the axial force σcbb2cos θ3D.
(a)
ð11Þ
(b)
!2 pffiffiffi 2 α tan θ3D sin θ3D cos θ3D 2
V cu1‐3D ¼ σ cb dc 1−
ð12Þ
where α is the aspect ratio of the panel zone, defined as:
α¼
h : dc
ð13Þ
From Eq. (12) it can be seen that the inclination angle θ3D should be determined first, in order to calculate the shear capacity of main compression strut (Vcu1-3D). Based on the lower bound theorem of classical plasticity theory [10], the stress state illustrated in Fig. 3 is a statically admissible field and the corresponding shear capacity is the lower bound of the true shear capacity. The maximum value of Vcu1-3D with respect to θ3D in Eq. (12) represents the closest estimation of the true shear capacity. Therefore, the inclination angle θ3D can be determined when Vcu1-3D reaches its maximum value in Eq. (12). In the meantime, θ3D must satisfy the geometrical constraints of the main strut given in pffiffiffi Eq. (14), i.e. tanθ3D should be less than 2 dc/h otherwise point C1 will exceed the bounds of concrete core (see Fig. 3).
0b tan θ3D b
pffiffiffi 2 α
ð14Þ
(c)
Fig. 7. Derivation of the shear capacity of the additional compression strut for a 3D joint (a) horizontal force equilibrium, (b) ultimate limit state of steel flanges, and (c) axial force equilibrium of additional strut.
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To find the maximum point of Vcu1-3D, Eq. (12) is differentiated with respect to θ3D. Eq. (15) can be obtained therefore:
ð15Þ Eq. (15) is a nonlinear equation that determines the relationship between the aspect ratio α and inclination angle θ3D. It is difficult to find the analytical solution of inclination angle θ3D as a function of aspect ratio α. Matlab 2009b® was used to obtain a numerical solution for various values of aspect ratio α in the range of engineering practice (from 0.5 to 3) as shown in Fig. 4. As shown in Fig. 4, the inclination angle θ3D decreases with the increase of aspect ratio α in the range from 0.5 to 3. In this range, a simplified formulation for θ3D as a function of α is obtained by data regression analysis as: θ3D
dc 1 : ¼ 0:468 arctan ¼ 0:468 arctan α h
ð16Þ
The simplified formulation in Eq. (16) agrees well with the numerical results based on Eq. (15) as depicted in Fig. 4. Furthermore, the θ2D versus α curve was also derived for planar shear states by Fukumoto and Morita [5] and is illustrated in Fig. 4. It can be seen from Fig. 4 that, for a certain aspect ratio, the inclination angle in the planar shear state (θ2D) is always greater than that in the 3D shear state (θ3D). The shear capacity of the main compression strut can then be rewritten as: V cu1‐3D ¼
1 2 σ b sin2θ3D 2 cb
ð17Þ
where b and θ3D are determined by Eqs. (11) and (16) respectively. The x or y component (see Fig. 3) of shear capacity of the main compression strut in a 3D state can be calculated as: V cu1‐x ¼ V cu1‐y
pffiffiffi pffiffiffi 2 2 2 V cu1‐3D ¼ σ b sin2 θ3D : ¼ 2 4 cb
ð18Þ
The shear capacities under 3D and planar loading states (calculated according to Eqs. (17) to (18)) are illustrated in Fig. 5. It is indicated that the x component of shear capacity in a 3D shear state is always smaller than that in a planar shear state except for very large aspect ratios. This means that the spatial coupling effect weakens the shear
Vcu-x/Vcu-2D
pffiffiffi 2 " pffiffiffi ! # α tan θ3D −1 pffiffiffi 2 ∂V cu1‐3D α sin θ3D − cos θ3D cos2 θ3D þ 2α sin θ3D ¼ 0: ¼ 2 2 cos θ3D ∂θ3D
1.0
0.8
rw =20 rw =40
0.6
rw =60 rw =80
0.4
0
1
2
3
Aspect Ratio α =h/dc Fig. 9. Relation between rw and Vcu-x/Vcu-2D for rσ = 10.
capacity of the main compression strut in one plane, and this reduction becomes more notable for smaller aspect ratios. In a similar manner to the additional compression strut mechanism in a planar shear state [5], the additional compression struts in a 3D shear state are illustrated in Fig. 6. Four additional compression struts exist, as shown in Fig. 6 (with different colors), and they are confined by the flanges of the steel tube. Due to the interaction of the steel tube flange, two plastic hinges may be formed at the two ends of each additional compression strut. The formulation of the shear capacity Vcu2-3D of one additional strut is derived according to the mechanism illustrated in Fig. 7(a), (b), and (c). Because the x or y components of Vcu2-3D are provided by two additional compression struts, the contribution of each additional strut to pffiffiffi the overall shear capacity is obtained as 2Vcu2-3D/4. Considering the horizontal force equilibrium of each additional strut as shown in Fig. 7(a), Eq. (19) is established: pffiffiffi 2 V cu2‐3D ¼ pbla 4
ð19Þ
where p is the pressure from the flange of the steel tube (see Fig. 7(a)); b is a dimension parameter determined by geometric relation (Eq. (11)); la is the distance between two plastic hinges formed within the steel tube flanges and can be determined by following Eq. (20): 1 2 pbla dψ ¼ 2M f dψ 2
ð20Þ
0.60
1.0
Ratio of Vcu2-x andVcu2-2D Ratio
Vcu-x/Vcu-2D
0.58
0.56
rσ=5
0.8
rσ=10 rσ=15
0.6
rσ=20 0.4
0.54 0.5
1.0
1.5
2.0
2.5
3.0
Aspect Ratio α Fig. 8. Ratio of x (or y) component of 3D shear capacity to planar shear capacity of an additional strut as a function of aspect ratio α.
0
1
2
Aspect Ratio α =h/dc
Fig. 10. Relation between rσ and Vcu-x/Vcu-2D for rw = 40.
3
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247
V3D-x/V2D
1.0
rw=20 rw=40
0.8
rw=60 rw=80 0.6
0
1
2
3 Fig. 13. Shear–deformation relationship with four linear segments for the panel zone.
Aspect Ratio α =h/dc Fig. 11. Relationship between rw and V3D-x/V2D for rσ = 10.
where dΨ is an arbitrary small virtual rotational angle; Mf is the plastic moment capacity of the flange plate of the steel tube and can be expressed as: Mf ¼
1 2 bt f 4 s sy
ð21Þ
where b is a dimension parameter determined by geometric relation (Eq. (11)); ts is the thickness of the plate of the steel tube flange; and fsy is the material yield strength of the steel tube. Eq. (20) was derived with consideration of the ultimate limit state of the steel tube flange (see Fig. 7(b)). The virtual work of the pressure from the additional strut should equal to the virtual work of the plastic hinges of the steel flanges because of the principle of virtual work. In the ultimate limit state, the compression stress in the additional strut reaches the compression strength of concrete (see Fig. 7(c)), and Eq. (22) can therefore be obtained as: σ cb cos2 ϕ ¼ p
ð22Þ
where ϕ is the dihedral angle between the plane of the steel tube flange and the plane of the cross section of the additional strut (see Fig. 7(c)). ϕ is determined by the geometric relationship as: cos2 ϕ ¼
1 sin2 θ3D 2
ð23Þ
V3D-x/V2D
1.0
rσ=5 rσ=10
0.8
rσ=15
where the inclination angle θ3D of the main strut can be calculated by Eq. (16). Finally, the shear capacity of the additional strut in a 3D shear state is derived by combining Eqs. (19) to (23) as below: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V cu2‐3D ¼ 4 M f bσ cb sin θ3D :
ð24Þ
The x or y component (see Fig. 6) of shear capacity of the additional compression strut in a 3D state is actually: V cu2‐x ¼ V cu2‐y ¼
pffiffiffi pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V cu2‐3D ¼ 2 2 M f bσ cb sin θ3D : 2
ð25Þ
The shear capacity of the additional strut in a planar shear state takes the form of Eq. (26) [5]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V cu2‐2D ¼ 4 M0f dc σ cb sin θ2D
ð26Þ
where dc is the width of the panel zone; σcb is the compression strength of the concrete; θ2D is the inclination angle of the compression strut with respect to the square concrete core in a planar shear state; Mf′ is the plastic moment capacity of the flange plate in a planar shear state and has a similar form to Eq. (21) except that b is replaced by dc: M0f ¼
1 2 dc t f : 4 s sy
ð27Þ
Since the inclination angle (θ2D) in a planar shear state is always larger than that in a 3D shear state (θ3D), and dc is also always larger than b (see Eq. (11)), by comparing Eqs. (24) and (26) it can be concluded that the 3D shear capacity of the additional strut is always lower than the planar shear capacity, and the x or y component of its 3D shear capacity is even smaller than the shear capacity of the additional strut in a planar shear state. This indicates that the spatial coupling effect also weakens the shear capacity of the additional compression struts. The ratio of the x (or y) component of 3D shear capacity Vcu2-x to the planar shear capacity Vcu2-3D is shown in Fig. 8 as a function of the aspect ratio
rσ=20 0.6
Table 1 Coefficients C1 and C2 for calculation of yielding shear capacity Vpy for a 3D composite joint used in Eq. (33).
0
1
2
Aspect Ratio α =h/dc
Fig. 12. Relationship between rσ and V3D-x/V2D for rw = 40.
3
Relationship of shear strains
γpy
γpu
C1
C2
γsy b γcu ‐ x b γsu γcu ‐ x b γsy b γsu γsy b γsu b γcu ‐ x
γsy γcu ‐ x γsy
γsu γsu γcu ‐ x
1 γcu ‐ x/γsy 1
γsy/γcu ‐ x 1 γsy/γcu ‐ x
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rewritten as Eq. (29) using the steel to concrete strength ratio rσ and width to thickness ratio rw of the steel tube:
V cu‐x ¼
!2 !3 pffiffiffi2 pffiffiffi pffiffiffi pffiffiffiffiffi 2 rσ 24 2 2 2 1− α tan θ3D 5 sin θ3D σ cb dc 1− α tan θ3D cos θ3D þ 2 2 2 rw
ð29Þ
Fig. 14. Dimensions of the joint specimens (J202).
α. It can be seen from Fig. 8 that a larger respect ratio tends to induce a more significant spatial coupling effect and therefore causes more reduction in the shear capacity of the additional compression strut in a 3D shear state. Considering that shear capacity is often designed in the directions of the principal axes (x and y) of the two orthogonal beams, the shear capacity (Vcu-x or Vcu-y) of the concrete in the panel zone of a 3D composite joint can be calculated in these two directions according to Eq. (28), which was derived based on Eqs. (17) and (24):
V cu‐x ¼
sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffi pffiffiffi 2 2 Mf b cos θ3D þ 4 ðV cu1‐3D þ V cu2‐3D Þ ¼ bσ cb sin θ3D : 2 2 bσ cb ð28Þ
To investigate the effects of major design parameters on the reduction of shear capacity due to the spatial coupling effect, Eq. (28) is
where the steel to concrete strength ratio rσ equals fsy/σcb and the width to thickness ratio of the steel tube rw equals dc/ts. It can be seen that for specified σcbd2c , the aspect ratio α, the strength ratio rσ and the width to thickness ratio of the steel tube rw are the factors that influence the shear capacity of the concrete panel. Figs. 9 and 10 show the influence of the aspect ratio (α) on the ratio of the shear capacity of the concrete core in a 3D loading state and in a 2D loading state (Vcu-x/Vcu-2D) with different width to thickness ratios (rw) and strength ratios (rσ). Both figures show that the aspect ratio α has remarkable influence on Vcu-x/Vcu-2D and a smaller aspect ratio α corresponds to a more significant reduction in the value of Vcu-x/Vcu-2D due to the spatial coupling effect. The shear capacity of the concrete core in a 3D loading state is only 50% of that in a 2D loading state with a small aspect ratio (0.5 b α b 1). This means that the spatial coupling effect cannot be ignored for joints with small aspect ratios. Figs. 9 and 10 also show that the 3D shear capacity of the concrete core (Vcu-x ) decreases with a decrease in the width to thickness ratio or an increase in the strength ratio, although the influences of these two factors are not as significant as that of the aspect ratio. The deformation at the ultimate shear strength of a 3D concrete core is given by Fukumoto and Morita [5]:
γcu‐x ¼ κ c
V cu‐x α cu Ac Gc
ð30Þ
where κc is the shear coefficient of the concrete core (1.2 for square concrete); Gc is the shear modulus of the concrete core; Ac is the
Fig. 15. Schematic diagram of experimental setup, joint details, and loading configuration for specimen J202.
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251 Table 2 Summary of four joint specimens. Specimen no.
Loading
Concrete slab
Concrete encased in steel tube
Steel beams
J101 J202 J203 J301
Bi-directional Bi-directional Uni-directional Bi-directional
No Yes Yes Yes
Yes Yes Yes No
In two planes In two planes In one plane In two planes
cross-sectional area of the concrete core. αcu is the stiffness reduction ratio as given by Fukumoto and Morita [5]: α cu ¼ 0:00158σ cb þ 0:0411
h þ 0:086 dc
ð31Þ
where the compression strength of the concrete σcb is in units of N/mm. 2.3. Shear capacity and deformation of a 3D composite joint The total shear capacity V3D-x of a joint in a 3D loading state is the superposition of the shear capacity of the concrete core (Eq. (6)) and that of the steel webs (Eq. (28)):
V 3D‐x ¼ V su þ V cu‐x
sffiffiffiffiffiffiffiffiffiffiffi! pffiffiffi 2 Mf b cos θ3D þ 4 ¼ bσ cb sin θ3D 2 bσ cb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Aw f su −σ 2s pffiffiffi þ : 3
ð32Þ
It is also important to evaluate the ratio of shear capacities in 3D and planar loading states (V3D-x/V2D). For such evaluation, a yield to tensile ratio (fsu/fsy) of 1.3 and an axial compression ratio (N0/(Acσcb + Asfsy), where N0 is column axial force, Ac and As are section areas of concrete and steel tube respectively) of 0.2 are adopted to calculate the joint shear capacity V3D-x (using Eq. (32)). Figs. 11 and 12 clearly illustrate the effects of the aspect ratio α, width to thickness ratio rw and strength ratio rσ on the reduction of joint shear capacity subjected to loading from 2D to 3D. Again, the aspect ratio shows significant reduction in capacity. Also, higher width to thickness ratios and lower strength ratios lead to more remarkable spatial coupling effects. The effects of the width to thickness ratio rw and the strength ratio rσ are more considerable on joint shear capacity with smaller aspect ratios. In the worst case (i.e., small aspect and strength ratios but a large width to thickness
249
ratio), joint shear capacity in 3D loading may correspond to only 60% of that in 2D loading. The shear–deformation relationship of a 3D composite joint may include four linear segments, as a result of the superposition of the trilinear model of the steel tube and the ideal elastoplastic model of encased concrete. This is shown in Fig. 13 where point A (with a shear deformation value of γsy, see Eq. (7)) corresponds to the yielding point of the panel zone; point B (with a shear deformation value of γcu-x, see Eq. (30)) is determined by the shear deformation at which the concrete core just reaches the ultimate shear strength; and point C (with a shear deformation value of γsu, see Eq. (8)) represents the ultimate shear strength of the panel zone corresponding to the ultimate state of the steel tube. Therefore, the total yielding (Vpy) and the ultimate (Vpu) shear capacity of a composite joint can be defined as: V py ¼ C 1 V sy þ C 2 V cu
ð33Þ
V pu ¼ V su þ V cu
ð34Þ
where Vsy and Vsu are the yielding and ultimate shear capacities of the steel tube in the joint region; Vcu-x is the ultimate shear capacity of the concrete in the panel zone of the joint; C1 and C2 are the coefficients that can be determined according to Table 1. In this table, γsy and γsu are the yielding and ultimate shear strains of the steel tube of a 3D joint (see Eqs. (7) and (8)); γcu-x is the ultimate shear strain of the concrete in the panel zone of a 3D joint (see Eq. (30)); γpy and γpu are the calculated yielding and ultimate shear strains of the joint region. For composite joints with fully developed shear strength and shear strain in the joint region, the webs of the steel tube yield first and then the concrete in the panel zone reaches its ultimate shear strain. Finally, the web of the steel tube reaches its ultimate state. For such joints, the relationship γsy b γcu-x b γsu is valid, i.e. C1 and C2 in Eq. (33) are determined according to the second row in Table 1. 3. Experimental validation To validate the analytical modeling developed above, four specimens (J101, J202, J203 and J301) were designed and tested under reversedcyclic loading [7]. The specimens were CFT columns with steel beams through interior diaphragm connections. Specimen J202 was the reference specimen, the dimensions of which are shown in Fig. 14 in detail. The geometries of specimens J101 and J301 were the same as that of J202, but no concrete slab was placed on top of the steel beam for J101, and no encased concrete was placed in the steel tubular column (but with a different tube thickness of 12 mm) for J301. Specimen
Fig. 16. Typical failure modes of the four specimens (a) fracture of steel tube (J101); (b) crushing of encased concrete (J202) and (c) overall deformation (J101).
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Table 3 Comparison of predicted shear capacities with experimental results. No.
Vsu/kN
Vcu/kN
Vpu/kN
Veu/kN
J101 J202 J203 R1 R6
1336.0 1336.0 1336.0 806.4 808.7
284.0 275.4 457.2 1579.6 914.8
1620.0 1611.4 1793.2 2386.0 1723.5
1743.7 1594.8 1944.0 2384.6 1787.6
J203 had beams in only one plane and was tested under a uni-directional reversal loading whereas the other specimens had beams in two planes and were tested under bi-directional reversal loading. The experimental setup and loading configuration are shown in Fig. 15 and the key experimental parameters of the different joint specimens are listed in Table 2. Steel plates of 8 mm, 10 mm and 12 mm thickness were used to make the steel species; their measured yielding strengths fsy were 381.4 MPa, 345.9 MPa and 388.4 MPa, respectively. The measured compressive strengths (σcb) of the encased concrete in the columns and the concrete slabs were 51.1 MPa and 36.4 MPa respectively. During the tests, a constant axial force (2458 kN for J101, J202 and J301 and 2116 kN for J301) was first applied on the top of the column as shown in Figs. 14 and 15. Vertical reversed-cyclic loads were applied at the ends of four beams (two beams in each plane), in a force control mode until the measured strains in the steel beam or steel panel exceeded the material yield strain, and subsequently in a displacement control mode. The final failure mode of all four specimens was a typical shear failure in the panel zone of the joints, as shown in Fig. 16. The shear failures were characterized by fracture of the steel wall in the panel zone and crushing of the encased concrete within. These specimens therefore provide valuable information for examining the proposed analytical model developed for the joint shear capacity. According to the loading configuration of the specimens, the shear force Vj in the joint region could be calculated by Eq. (35) [11]: 2Q b Lb 2Q b Lb Q L −Q c ¼ − b H H b −t bf Hb −t bf
ð35Þ
Shear Force V/kN
where Qc is the vertical loading at the end of the beams and Qc is the horizontal reaction force at the top of the column; Hb is the height of the beam; tbf is the flange thickness of the beam; Lb is the distance from the loading point of the beam to the flange of the column; H is the joint height; L is the distance between the two loading points of the beams in the same plane (see Fig. 14). The shear capacities of the specimens were calculated using the proposed formulation developed for 3D composite joints (see Eq. (32)). The 2500
2500
2000
2000
1500
1500
1000 500
-3
γ /10
0 -40
-20
-500
0
20
40
-1000
experimental predicted
-1500 -2000 -2500
Shear Force V/kN
Vj ¼
calculated values are compared with the experimental results in Table 3, where Vsu and Vcu are the calculated contributions from the steel tube and the encased concrete according to Eqs. (6) and (28) respectively; Vpu is the total shear capacity of the 3D joint calculated by Eq. (32) and Veu is the measured shear capacity of the joint from experiments. Overall, good agreement between the calculated (Vpu) and measured (Veu) total shear capacities was found, with a maximum discrepancy less than 8%. Specimen J202 associated with beams in two planes was also compared to J203 with beams only in one plane. It was found that that the experimental result for the joint shear capacity of J202 was about 18% lower than that of J203, and 11% lower than the predicted shear capacity of J203. This comparison indicates that using the planar shear formulation may overestimate the joint shear capacity in the case of a 3D loading configuration and this may lead to unsafety. Further, joint shear–deformation curves were formed by the calculated shear capacity values (Vsy, Vsu and Vcu-x, according to Eqs. (5), (6) and (28)) and the corresponding shear deformation (γsy, γsu and γcu-x) according to Eqs. (7), (8) and (30) for specimen J202, as shown in Fig. 17. The highly nonlinear experimental shear–deformation skeleton curve can be relatively well described by the proposed modeling curve with four linear segments. Discrepancies between the experimental and predicted shear–deformation skeleton curves were mainly found at the second and third segments between the yield and ultimate states (see Fig. 13). These discrepancies may resulted from the assumption of an ideal elastoplastic shear–deformation relationship of the concrete core in Fig. 13. The 3D shear capacity model was further validated by the experimental results reported by Nishiyama et al. [4]. The specimens R1 and R6 [4] were tested under uni-directional and bi-directional reversal loading respectively, and were similar to specimens J202 and J101 investigated by [7]. The shear capacities of R1 and R6 were calculated using the shear capacity formulations developed in this paper and considering the material and dimensional parameters provided by Nishiyama et al. [4]. The experimental shear capacity was derived by Eq. (35) based on the force Qc (i.e. column shear force) applied at the end of the column [4]. The predicted and experimental results are compared in the last two rows in Table 3 and good agreement is evident, with discrepancies less than 5%. The planar shear capacity of specimen R6 was calculated as 2222.6 kN by the equations introduced by Nishiyama et al. [4] and this value was 24% higher than the experimental result (1787.6 kN, see Table 3). It is evident for specimen R6 that the spatial coupling effect significantly weakened the shear capacity of the joint subjected to 3D loading. This is because in this case, highstrength concrete (σcb = 97.7 MPa) was used for specimen R6, leading to a rather small steel to concrete strength ratio (rσ = fsy/σcb = 5.0). Together with the other factors (α = h/dc = 1.0, and rw = dc/ts = 55.6 according to Eq. (29)), this likely caused the more remarkable reduction to the shear capacity of the 3D joint.
1000 500
-3
γ /10
0 -40
-20
-500
0
-1000
20
40
experimental predicted
-1500 -2000 -2500
(a)
(b)
Fig. 17. Comparison of experimental and predicted skeleton curves of specimen (a) J202 (bi-directional loading) and (b) J203 (uni-directional loading).
J. Fan et al. / Journal of Constructional Steel Research 112 (2015) 242–251
4. Conclusions An analytical model was developed in this paper to estimate the shear capacity of 3D composite joints of CFT columns and steel beams subjected to shear forces in two planes. In this model, the main compression strut and additional compression strut mechanisms were formulated for the concrete core of the joint panel subjected to 3D loading scenario. The joint shear capacity was calculated and compared with experimental results where the specimens were subjected to reversedcyclic loading in two planes and failed through shear mode at the joints. The joint shear–deformation relationship was characterized by a piecewise linear model with four segments, taking into consideration the effects of shear forces in two planes. The modeling results of joint shear capacities and shear–deformation relationships were validated by the experimental results from references. Based on this work, the following conclusions can be drawn: 1) The analytical model developed in this paper suggests that 3D (bidirectional) loading weakens the shear capacity of the concrete panel zone, although it does not affect the shear capacity of the webs of the steel tube. Because of the loadings from two planes and the associated space coupling effects on a 3D joint, the main and additional compression struts, obtained as a result of superposition of two 2D joints in the corresponding plane loading state, may present different orientations, shapes and sizes from those in a planar loading state. 2) A new piecewise linear constitutive model based on the superposition principle was implemented into the shear–deformation relationship of 3D composite joints. This constitutive model was characterized by the analytic ultimate shear strength formula in 3D shearing states, and therefore the spatial coupling effect caused by bi-directional shear to the concrete zone could be taken into consideration. The predicted and experimental shear–deformation curves showed satisfactory agreement. The major discrepancy was found between yield and ultimate states because of the assumption of an ideal elastoplastic shear–deformation relationship of the concrete core. 3) The proposed model for the shear capacity of composite joints of CFT columns was validated by the results from 2D and 3D joint experiments. Further calculations also indicated that using the planar shear capacity formulation for a joint in a 3D shear state may lead to an overestimation of 10%–20% for the joint shear capacity. The reduction in joint shear capacity due to a 3D loading configuration depends mainly on the aspect ratio of the joint panel, the steel to
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concrete strength ratio and the width to thickness ratio of the steel tube. A decrease in the aspect ratio results in the most notable reduction in joint shear capacity and such a reduction also becomes more considerable with an increase in the width to thickness ratio or a decrease in the strength ratio. For a worst-case scenario where the joint has a relatively small aspect ratio (0.5 for example), a large width to thickness ratio (80.0 for example) and a small strength ratio (5.0 for example), joint shear capacity in the 3D shear state can reduce to only 60% of that in the 2D loading state.
Acknowledgment The authors appreciate the financial support of the National Natural Science Foundation of China (No. 51222810) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120002110003). References [1] B.D. Koester, Panel Zone Behavior of Moment Connections Between Rectangular Concrete-filled Steel Tubes and Wide Flange Beams, University of Texas at Austin, 2000. [2] C.-T. Cheng, L.-L. Chung, Seismic performance of steel beams to concrete-filled steel tubular column connections, J. Constr. Steel Res. 59 (2003) 405–426. [3] C.-T. Cheng, C.-F. Chan, L.-L. Chung, Seismic behavior of steel beams and CFT column moment-resisting connections with floor slabs, J. Constr. Steel Res. 63 (2007) 1479–1493. [4] I. Nishiyama, T. Fujimoto, T. Fukumoto, K. Yoshioka, Inelastic force–deformation response of joint shear panels in beam-column moment connections to concrete-filled tubes, J. Struct. Eng. 130 (2004) 244–252. [5] T. Fukumoto, K. Morita, Elastoplastic behavior of panel zone in steel beam-toconcrete filled steel tube column moment connections, J. Struct. Eng. 131 (2005) 1841–1853. [6] Architectural Institute of Japan (AIJ), AIJ Standard for Structural Calculation of Steel Reinforced Concrete Structures, 1987. (Tokyo, Japan). [7] J.S. Fan, Q.W. Li, J.G. Nie, H. Zhou, Experimental Study on the seismic performance of 3D joint between concrete-filled square steel tubular column and composite beams, J. Struct. Eng. (2013)http://dx.doi.org/10.1061/(ASCE)ST.1943-541X.0001013 (accepted for publication). [8] J.G. Nie, K. Qin, C.S. Cai, Seismic behavior of connections composed of CFSSTCs and steel–concrete composite beams—finite element analysis, J. Constr. Steel Res. 64 (2008) 680–688. [9] T. Paulay, R. Park, M.J.N. Preistley, Reinforced concrete beam-column joints under seismic actions, ACI Journal Proceedings, ACI, 1978. [10] W.-F. Chen, Limit Analysis and Soil Plasticity, Elsevier, 2013. [11] S. Sasaki, M. Teraoka, K. Morita, T. Fujiwara, Structural behavior of concrete-filled square tubular column with partial-penetration weld corner seam to steel Hbeam connections, Proceedings of the 4th Pacific Structural Steel Conference, 2 1995, pp. 33–40.